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Vortex-induced vibration of flexible pipe fitted with helical strakes in oscillatory flow
Ocean Engineering 189 (2019) 106274
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Vortex-induced vibration of flexible pipe fitted with helical strakes in oscillatory flow Haojie Ren a, b, Yuwang Xu a, b, *, Jingyun Cheng c, Peimin Cao c, Mengmeng Zhang a, b, Shixiao Fu a, b, Ziyang Zhu a a b c
State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai, China Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration, Shanghai, China SBM Offshore, Houston, TX, USA
A R T I C L E I N F O
A B S T R A C T
Keywords: Vortex-induced vibration Helical strake Flexible pipe Oscillatory flow
Helical strakes are widely used to suppress vortex-induced vibration (VIV) in offshore engineering. However, the VIV response and suppression efficiency of the straked pipe remain unclear in an oscillatory flow. In this paper, an experimental study of VIV for a flexible pipe fitted with helical strakes was conducted in an oscillatory flow with the Keulegan-Carpenter (KC) number varying from 21 to 165 and maximum reduced velocities ranging from 4 to 12. The effects of the helical strakes on the VIV response, Strouhal number, suppression efficiency and fatigue damage in an oscillatory flow were investigated. The results show that the suppression efficiency and fatigue damage reduction ratio are not as ideal in oscillatory flow as those in steady flow. Moreover, the VIV dominant frequency is obviously magnified and a large Strouhal number is observed, reaching 0.439 under a lower reduced velocity of 4. As the reduced velocity further increases to 6, 8, 12, two branches of Strouhal numbers are clearly seen. The dominant frequency decreases at the first branch with a small KC number and increases at the second branch with a large KC number.
1. Introduction As oil and gas production moves into deeper waters, risers are becoming increasingly slender. Under the action of ocean currents, vortices are generated and alternately shed from the sides of these flexible slender risers, resulting in a periodic excitation force. If the frequency of the vortex-induced force is near the natural frequencies of the risers, a resonance vibration will occur. This is the so-called vortexinduced vibration (VIV) (Blevins and Saunders, 1977). This periodic vibration has been proven to be the main reason for fatigue damage of risers. Therefore, a means to effectively suppress VIV has been studied for decades in industry and academia. At present, there are two main methods to suppress VIV of the riser: active and passive control. The main difference between them is that active control requires additional power (He et al., 2000; Jeon et al., 2004; Williams and Zhao, 1989), while passive control does not. The latter is more widely used due to its easier manufacture. Different pas sive control methods to suppress VIV have been studied by many re searchers (Allen et al., 2004; Bearman and Owen, 1998; Galvao et al.,
2008; Gao et al., 2015; Trim et al., 2005). Helical strake is among the most widely used apparatuses in practical projects. Its high suppression efficiencies in the steady flow have been verified by both numerical simulations and experiment studies (Vandiver et al., 2006). tested three different distributions of strakes in the reduction of the VIV amplitude and frequency (Allen et al., 2004). presented the performance of different helical strake geometries at critical Reynolds numbers (Trim et al., 2005). studied the suppression effectiveness of the strake on a long flexible riser in both the CF and IL directions (Gao et al., 2015). pre sented the effects of strake configurations in both uniform and linearly sheared flows (Constantinides and Oakley, 2006; Pinto et al., 2006; Thiagarajan and Constantinides, 2005). studied the VIV suppression mechanism of helical strakes by using the computational fluid dynamic (CFD) method. They found that helical strake could prevent the inter action between two shear layers and break vortex shedding, thus sup pressing the VIV. The aforementioned studies focused on the performance of helical strakes under steady flow, and there are extremely few publications regarding the performance of helical strakes under oscillatory flow.
* Corresponding author. State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai, China. E-mail address: [email protected] (Y. Xu). https://doi.org/10.1016/j.oceaneng.2019.106274 Received 10 May 2019; Received in revised form 13 July 2019; Accepted 5 August 2019 Available online 26 August 2019 0029-8018/© 2019 Elsevier Ltd. All rights reserved.
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However, the marine risers, for instance, the steel catenary riser (SCR) and steel lazy wave riser (SLWR) act as the only channel to con nect the subsea wellhead and the platform. In a real sea state, the waveinduced periodic motion of the platform always results in a relatively equivalent oscillatory flow around the risers. It has been found from recent experimental study that the oscillatory flow induced by the pure motion of the platform can also excite the VIV at the sag-bend of the SCR (Wang et al., 2015, 2017a), termed vessel motion-induced VIV (VMI- VIV) (Pesce et al., 2017). The VMI-VIV can cause serious fatigue damage to risers (Wang et al., 2014). To improve the understanding of this phenomenon (Fu et al., 2014), conducted a flexible pipe model test in oscillatory flow by forcing the model to oscillate in still water with different periods and amplitudes. A VIV developing process in oscilla tory flow was first proposed, including building up, lock-in and dying out. This novel VIV phenomenon has attracted much attention from both industry and academia. Another experiment with a SLWR was presented by (Cheng et al., 2016) and (Constantinides et al., 2016). The similar VMI-VIV phenomenon was also observed. To suppress the VIV induced by vessel motion, tests of a pipe model fitted with helical strakes with a pitch of 15 D and a height of 0.25 D were further conducted. However, VIV still occurred for the straked pipe, which is contrary to the popular belief that helical strakes can nearly fully suppress VIV. What causes this result is still unclear. To further improve the understanding of VIV performance for straked pipes in the equivalent oscillatory flow, we conducted VIV model tests for a straked pipe under oscillatory flow in SKLOE. A straked pipe with pitch of 15 D and height of 0.25 D was forced to harmonically oscillate under various combinations of amplitude and period with Keulegan-Carpenter (KC) numbers varying between 21 and 165 and maximum reduced velocities ranging from 4 to 12. The effects of the helical strakes on VIV response, Strouhal number, suppression efficiency and fatigue damage are discussed.
Fig. 2. Simplified sketch of the setup.
2. Model test
Fig. 3. Detailed view of the end connection.
2.1. Test apparatus
random (PPR) pipe outside and a copper cable inside to satisfy the designed mass and bending stiffness. Silicone gel was placed between different layers to avoid relative slippage. As shown in Fig. 4, the height, pitch and number of starts of the helical strake were three of its main geometric parameters, which were 0.25D, 15D and 3 in the experi mental study, respectively. The helical strakes were tied to the flexible pipe. The main properties of the test model are listed in Table 1. The 1st and 2nd order eigen frequency of the flexible pipe in still water can be calculated by Eq. (1) and were 1.77 Hz and 3.86 Hz, respectively.
A model test was conducted in an ocean basin at Shanghai Jiao Tong University. The entire experimental setup, primarily containing two horizontal and vertical tracks and a flexible pipe model, was installed under the bottom of a sub-carriage as shown in Fig. 1 and Fig. 2. Details of the connection frame at the end side of the experimental apparatus are shown in Fig. 3. The pipe model was connected to a force sensor through a universal joint. The force sensor was connected to a tensioner that was fixed to the slider of the vertical track. A pretension force of 500N was applied to the flexible pipe. Meanwhile, two circular end plates were installed to avoid the perturbation caused by the motion of the supporting frame. The flexible pipe model was constructed with a polypropylene
Fig. 1. Overview of the whole experimental setup.
Fig. 4. Geometric parameters of the helical strakes. 2
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natural mode is the potential excitation mode for all the test cases. Notably, the added mass may deviate from the value of 1.0 in still water, and the forced oscillation periods will change with the KC number at the same maximum reduced velocity. All the test cases were divided into four groups with different maximum reduced velocities, as presented in Table 2. The correspond ing maximum Reynolds number (Re) has been also list in this table, which can be calculated by Eq. (6). Each test group had test cases with a KC number ranging from 21 to 165 under the same amplitude of the forced motion velocity. The same cases were also conducted for a bare pipe.
Table 1 Parameters of the flexible pipe model. Item
Value
Model length L (m) Outer diameter D (mm) Mass in air m (kg/m)
4 29 1.696
Bending stiffness EI (N⋅m2) Tensile stiffness EA (N) Pre-tension T (N) Damping ratio ζ (%) Calculated first natural frequency f10 in water (Hz) Calculated second natural frequency f20 in water (Hz)
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n T0 n2 π2 EI 1 fn0 ¼ þ 2 ⋅ ; m ¼ m þ πD2 ρCm ðCm ¼ 1:0Þ 2L m 4 L m
46.433 1.528 Eþ06 500 2.53 1.77 3.86
Remax ¼ n ¼ 1; 2::::
3. Basic theory 3.1. Preprocessing For convenience in description, the coordinate system is defined as O-XYZ: the origin is at one of the ending points of the riser and the Zaxis, X-axis and Y-axis are along the axis of the riser, IL direction and CF direction, respectively, as shown in Fig. 2. At the start of the experiment, a constant pre-tension of 500N was exerted at the flexible pipe ends. However, the pretension periodically varied with pipe oscillation. Thus, the measured strain included three components: the initial axial strain εT0 caused by pre-tension, the vari able axial strain εT(t) caused by varying tension and the bending strain εVIV caused by the vortex-induced vibration. The original measured strains εCF_a (z,t) and εCF_c (z,t) sampled by CF_a and CF_c measurement points, respectively can be correspondingly expressed as,
2.2. Test matrix To investigate the VIV response characteristics of the flexible pipe fitted with helical strakes in oscillatory flow, the model was forced to oscillate in harmonic motion under various combinations of amplitude and period in the horizontal direction. The instantaneous displacement X(t) and velocity of the forced motion U(t) can be expressed as follows: � � 2π XðtÞ ¼ Am sin t (2) T
VR ¼
εCF a ðz; tÞ ¼ εT0 þ εT ðz; tÞ þ εVIV ðz; tÞ
(7)
εCF c ðz; tÞ ¼ εT0 þ εT ðz; tÞ
(8)
(3)
εVIV ðz; tÞ ¼ ½εCF a ðz; tÞ
2πAm D
(4)
2π⋅Am T⋅f10 ⋅D
(5)
εVIV ðz; tÞ
Combing Eqs. (7) and (8), the pure VIV strain at z location εVIV (z,t) can be calculated by
where Am and T are oscillation amplitude and period, respectively. The KC number and maximum reduced velocity VR are considered two of the main parameters that determine the VIV characteristics of a flexible cylinder under oscillatory flow and are represented as: KC ¼
(6)
υ
where Um is the forced motion velocity amplitude. υ is the kinematic viscosity coefficient. In our experiment, the ambient temperature is maintained near 15 � C and υ is therefore approximately 1.14 � 10 6 m2 s 1.
(1)
where T0 is the pretension force. m and m are the mass per unit length in still water and air, respectively. L and D are the length and diameter of flexible pipe, respectively. EI is the bending stiffness of pipe. ρ is the density of the water, ρ ¼ 1000 kg/m3. The added mass coefficient Cm is assumed to be 1.0. It should be noted that the diameter values are all kept as that of bare pipe. Four groups of Fiber Bragg Grating (FBG) strain sensors were installed to measure the strain responses in both the CF and IL directions. Each of the FBG groups (CF_a, CF_c, IL_b and IL_d) had ten measurement points along the pipe separated by 0.36 m, as shown in the schematic diagram in Fig. 5.
� � 2π 2π UðtÞ ¼ Am cos t T T
Um D
(9)
εCF c ðz; tÞ�=2
Fig. 6 shows the original VIV strain time history of VR ¼ 6 and KC ¼ 143. The original VIV strain obviously contains significant fo and Table 2 Test matrix.
where f10 denotes the first natural frequency of the test model in still water. f10 is usually used to calculate the reduced velocity since the first
Case No.
VR
Am(m)
KC
Um (m/s)
Remax
1–19 20–40 41–61 62–82
4 6 8 12
[0.10:0.02:0.76] [0.10:0.02:0.76] [0.10:0.02:0.76] [0.10:0.02:0.76]
21–165 21–165 21–165 21–165
0.206 0.309 0.411 0.617
5240 7860 10455 15696
Fig. 5. FBG strain sensor instrumentation along the flexible pipe model. 3
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Fig. 6. Time history of the original VIV strain at VR ¼ 6 and KC ¼ 143.
2fo frequency components, which is caused by the swing of the flexible pipe. fo refers to the forced oscillation frequency and is equal to 1/T. As Fig. 7 (a) shows, the flexible pipe has an initial deflection under the action of gravity. Combined with the forced motion, the flexible pipe with the initial deflection performs a pendulum movement. To further verify this view, Fig. 7(b) shows the trajectory of the strain at the 5th gauge points. The blue solid line represents the original total strain including the VIV strain and bending strain caused by the pendulum movement. The red solid line is these bending strain components below 2.5 fo after low-pass filtering. Through Fig. 7(b), an obvious asymmet rical pendulum trajectory presented by red solid line can be observed. It also explains the significant f0 and 2f0 components that exist in the original VIV strains. This asymmetrical pendulum movement may be caused by the asymmetrical drag force in an oscillation period. To eliminate the effects of the flexible pipe pendulum motion caused by the forced motion and high frequency noise, a bandpass filter was utilized. The passband cutoff frequencies were set as 2.5fo and 30 Hz. A comparison of the original VIV strain and filtered VIV strain at the 5th gauge points (VR ¼ 6 and KC ¼ 143) is shown in Fig. 8. Although the strain caused by the cycloidal motion plays an important role in bending strain response, this strain cannot be counted in the VIV strain because this strain is not caused by the vortex shedding.
n X
pi ðtÞϕi ðzÞ;
yðz; tÞ ¼
(10)
z 2 ½0; L�
i¼1
where y(z,t) is the VIV displacement response in the CF direction at the z location, and pi(t) is the ith generalized coordinate displacement values at time t; ϕi(z) is ith displacement modal shape at the z location. Based on the small deformation assumption, the curvature κ(z,t) can be expressed as, κðz; tÞ ¼
n ∂2 yðz; tÞ X ¼ pi ðtÞϕ00 i ðzÞ; 2 ∂z i¼1
z 2 ½0; L�
(11)
where ϕ’’i is the ith modal shape of the curvature. According to the geometric relationship between the curvature and strain, the strain can be calculated by,
εðz; tÞ ¼ κðz; tÞR ¼ R
n X
pi ðtÞϕ00 i ðzÞ;
z 2 ½0; L�
(12)
i¼1
where R denotes the radius of the flexible pipe model at the z location. In our model, mode shapes of the displacement are sinusoidal, which can be expressed as,
3.2. Displacement identified method
ϕi ðzÞ ¼ sin
According to the Euler-Bernoulli beam theory, the VIV displacement response of the flexible pipe under the external load can be expressed as the sum of the mode shapes multiplied by the generalized coordinate values of each step. The VIV displacement response in the CF direction can be expressed as,
iπz ; L
i ¼ 1; 2:::
(13)
From Eq. (11), the mode-shapes of the curvature are also sinusoidal. After obtaining the mode shape of the displacement and curvature, the modal generalized coordinates can be obtained from Eq. (12). Then, the VIV displacement response in the CF direction can be calculated by Eq. (10). Fig. 9 presents the distribution of root mean square of VIV
Fig. 7. Experimental image and trajectory of strain at the 5th gauge points (VR ¼ 6 and KC ¼ 143). 4
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Fig. 8. Comparison of the original VIV strain and filtered strain in the CF direction at the 5th gauge points (VR ¼ 6 and KC ¼ 143).
Fig. 9. Distribution of root mean square of VIV displacement and VIV strain along the flexible pipe in the case of VR ¼ 6 and KC ¼ 30 (A ¼ 0.14m, T ¼ 2.86s): (a) Root mean square of VIV displacement; (b) Root mean square of VIV strain.
displacement (YRMS/D) and VIV strain (εRMS)along the flexible pipe in the case of VR ¼ 6 and KC ¼ 30. As shown in Fig. 9 (a), the blue and red lines present the VIV displacement of bare and straked pipe, respec tively. The VIV displacement of bare pipe is larger than that of straked pipe. It indicates that the helical strakes can also suppress the vortexinduced vibration in an oscillatory flow. To validate the displacement identification, the identified VIV strain was recalculated based on the reconstructed VIV displacement. Through Eq. (11), curvature can be obtained by the second order difference of VIV displacement to position z. Then identified strain values are further calculated by Eq. (12). As presented in Fig. 9 (b), the red dash line and blue solid line are the identified εRMS of straked and bare pipe, respec tively. The red circle and blue square symbols represents the measured εRMS of straked and bare pipe, respectively. The identified values are in a good agreement with the measured one. This consistency demonstrates the validity of displacement identification method for the flexible pipe in an oscillatory flow.
shedding frequency, the VIV characteristics in the CF direction vary with time. This “time-varying” feature has been reported to be the major difference between the VIV in steady flow and that in oscillatory flow (Fu et al., 2014). To investigate the time-varying feature of the flexible pipe fitted with helical strakes, the wavelet transformation is introduced to help analyze the time-frequency distribution of the VIV in oscillatory flow. The continuous wavelet transform equation is expressed as, � � � Z þ∞ t τ f ðtÞψ � ð dt (15) WTf a; τ ¼ ⟨ f ðtÞ; ψ a;τ ðtÞ⟩ ¼ a 1=2 a ∞ where WTf(a, τ) is the coefficient of the time domain signal after the wavelet transformation, which represents the frequency variation at that time scale. Parameter a is the scale factor, τ is the shift factor, and ψ (t) is the mother wavelet. In this paper, we use the Morlet wavelet equation as the mother wavelet, which can be defined as, �� 2 ψ t ¼ Ce t =2 cosð5tÞ (16)
3.3. Time-frequency analysis The shedding frequency changes with the periodic oscillation ve locity of the model, which can be written as: fst ðtÞ ¼
St⋅jUðtÞj D
3.4. Fatigue damage estimation The VIV can cause serious fatigue damage to marine risers. There fore, it is necessary to estimate the fatigue damage of the flexible pipe fitted with helical strakes under different maximum reduced velocities VR and KC numbers. The stresses σ at the outer pipe can be calculated by,
(14)
where St refers to the Strouhal number (typically, St ¼ 0.2); fst is the vortex shedding frequency. Under the effect of the periodically varying 5
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(17)
σ ðz; tÞ ¼ E⋅εðz; tÞ
where E is the elastic modulus of the flexible pipe. According to the Miner-Palmgren hypothesis, fatigue damage occurs when the accumulation of fatigue damage from the stress time histories with varying amplitudes satisfies Eq. (18) X ni Da ¼ ¼1 (18) Ni i where ni is the number of cycles with the stress range Δσ i; Ni is the number of cycles to failure for the specific stress range Δσi, which can be expressed as Eq. (19) according to the S–N curve. Ni ¼ 10ðloga
(19)
m logΔσi Þ
Thus, the total fatigue damage Dot can be calculated by the sum mation of the contribution from all cycles, Ntot X
Dot ¼
(20)
Dai i
where Ntot is the total number of the stress cycles in the VIV time history. The B1 S–N curve from (DNV-RP-C203 2011) is applied. The main parameters are listed in Table 3.
Fig. 10. Time history of the VIV response and time frequency distribution at the 5th gauge point of the bare pipe in the case of VR ¼ 6 and KC ¼ 30: (a) Timevarying frequency of the vortex shedding; (b) VIV displacement response; (c) Modal weight of displacement; (d) Time frequency distribution; (e) Timevarying natural frequency and dominant response frequency.
4. Results and discussions 4.1. VIV responses of the bare pipe in oscillatory flow To investigate the effects of helical strakes on VIV response in an oscillatory flow, an experimental study of a bare pipe was firstly con ducted as a benchmark. For the bare pipe test cases, the first mode is always dominant in our experiments. Therefore, the results measured at the 5th gauge point that is in the middle of the pipe model (z ¼ 1.73 m) is used for further analysis. Fig. 10 and Fig. 11 show the time history of the VIV response and time frequency distribution in the case of VR ¼ 6 under a small KC number (KC ¼ 30) and a large KC number (KC ¼ 121), respectively. Each figure contains five subplots. Figure (a) is the timevarying shedding frequency fst, which is calculated based on Eq. (14). Figure (b) is the VIV displacement response, which is reconstructed by utilizing the modal superposition method as proposed in Section 3.2. Figure (c) shows the time history of the modal weight p(t) of displace ment. Figure (d) is the wavelet analysis of the strain signal. The depth of the color indicates the concentration level of the strain components. Figure (e) presents the time varying natural frequency and VIV response dominant frequency. The blue dash and red solid lines represent the first and second order eigen frequency, while the black line refers to the dominant frequency. It should be noted that the first and second order eigen frequency (f1, f2) have been calculated based on the measured time-varying tension, assuming the added mass coefficient to be equal to 1. As shown in Fig. 10, the shedding frequency calculated by Eq. (14) varies from 0 to 2 Hz. The VIV displacement amplitude is less modulated and the maximum value reaches 0.64 D. As seen from the wavelet results shown in Fig. 10 (e), the dominant frequency is always close to f1. This indicates that the lock-in occurs at first mode. Although the response is dominated by the 1st natural mode, higher harmonics also occur under the small KC number (KC ¼ 30) presented in Fig. 10 (d). However, under the large KC number (KC ¼ 121) as shown in Fig. 11, a strong amplitude
Fig. 11. Time history of the VIV response and time frequency distribution at the 5th gauge point of the bare pipe in the case of VR ¼ 6 and KC ¼ 121: (a) Time-varying frequency of the vortex shedding; (b) VIV displacement response; (c) Modal weight of displacement; (d) Time frequency distribution; (e) Timevarying natural frequency and dominant response frequency.
modulation phenomenon appears and the VIV developing process is comprised of three phases: building up, lock-in and dying out. The maximum VIV displacement amplitude value reaches 0.65 D, which is the same as that in the small KC number case. These results are consis tent with the VIV features in oscillatory flow found by (Fu et al., 2014; Wang et al., 2014). Significant differences in the VIV response between the small and large KC numbers will provide a good reference for investigating the effects of helical strakes. The comparison of these distinct features between the bare and straked pipes will help in improving our understanding of the effects of helical strakes in an
Table 3 S–N curve parameters. S–N curve
loga
m
B1
14.917
4.0
6
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oscillatory flow.
Fig. 12(b) and(c). The response becomes very steady and is always dominated by the 1st natural mode, which is lower than the maximum shedding frequency. This finding indicates that the strakes can reduce the dominant frequency and suppress the higher frequency response components under higher reduced velocities, which is consistent with the results in uniform flow found by (Gao et al., 2016). The main reason for the difference between the low and high reduced velocities as previously described may be attributed to the competition between the effects of the helical strakes on the breaking vortex and the prevention of interaction between two shear layers. Breaking vortex effects means the helical strakes break the large orderly vortex into many weak and messy ones. The latter effect is that helical strakes prevent the two shear layers from affecting each other and make two layers parallel to the flow direction shown in Fig. 14. Under a lower flow velocity (a lower Re number), the vortex shedding from the two sides of the straked pipe is quite weak and very easily broken. The breaking vortex leads to a higher response frequency. However, the vortex be comes stronger and sheds from the two sides of the straked pipe in parallel under a higher flow velocity (a higher Re number). The inter action of the two shear layers is prevented by the helical strakes (Con stantinides and Oakley, 2006; Pinto et al., 2006). The response frequency is reduced under this condition. Only one effect works under steady flow, but two different effects on the response frequency will compete in oscillatory flow. For easier understanding, a sketch of the competition between the effects of the helical strakes under a small KC
4.2. VIV responses of the straked pipe in oscillatory flow 4.2.1. Time-varying responses of the straked pipe under a small KC number The effects of the helical strakes on the VIV response in an oscillatory flow with a small and large KC number were investigated by comparing them to the experimental results of the bare pipe model. Fig. 12 shows the VIV response and time frequency distribution at the 5th gauge points of the straked flexible pipe in the case of a small KC number (KC ¼ 38) with different reduced velocities ranging from 4 to 12. As shown in Fig. 12(a), an obvious “mode transition” phenomenon is observed under VR ¼ 4. The dominant frequency represented by the black solid line of the straked pipe changes back and forth between the 1st and 2nd natural frequencies. Assuming the St number is equal to 0.2, the maximum shedding frequency of the straked pipe based on Eq. (14) should be 1.42 Hz. However, this is much lower than the dominant response frequency found in the experiment, as shown in the third subplot in Fig. 12(a). This means that the helical strakes can result in a St number much larger than 0.2 (a typical value for a bare pipe) and significantly magnify the shedding frequency under a small reduced velocity. However, as the maximum reduced velocity increases to 6, 8 and 12, “mode transition” and higher harmonic components disappear, and the response characteristics are similar to those of the bare pipe, as shown in
Fig. 12. VIV response and time-frequency distribution at the 5th gauge point of the flexible pipe fitted with helical strakes at a small KC number: (a)VR ¼ 4; (b) VR ¼ 6; (c) VR ¼ 8; (d) VR ¼ 12. 7
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4.2.2. Time-varying responses of the straked pipe under a large KC number To further investigate the VIV responses under a large KC number, Fig. 15 shows the VIV response and time-frequency distribution at the 5th gauge point of the straked flexible pipe in the case of the large KC number, which is approximately 100. The reduced velocities vary from 4 to 12. As VR is equal to 4 shown in Fig. 15(a), a “mode transition” very often occurs, and the response frequency is quite unstable. A signifi cantly higher response frequency component can be observed. There are no obvious VIV processes under such a low reduced velocity, which is completely different from that of the bare pipe. The main reason is that the vortex-breaking effects of the helical strakes dominate. In the cases of a VR ¼ 6 and 8 shown in Fig. 15(b) and (c), respec tively, the time-frequency distribution becomes very stable. The response characteristics are similar to those of the bare pipes. A stronger amplitude modulation and obvious VIV developing process comprised of building up, lock-in and dying out can be seen. The vortex-breaking effects dominant under the low velocity cause higher harmonic re sponses in this case, differing from the bare pipe. As the reduced velocity increases to 12, a strong amplitude modu lation still occurs, while the VIV developing process and higher har monic response disappear. The shorter cycle of the forced motion and higher velocity may serve as the main reasons, which suppresses the vortex-breaking effects. In addition, the vortex-shedding process can be altered when the cylinder reverses its motion direction and encounters the previously shed vortices (Fu et al., 2014; Sumer and Fredsøe, 1988). The interaction stops the VIV from developing and leads to a stronger response as previously explained. The maximum values of the displacement at VR ¼ 6, 8 and 12 are 0.07D, 0.13D and 0.21D, respec tively. The displacement of VR ¼ 12 is obviously larger than that at VR ¼ 6 and 8. This is consistent with the previously explained reason.
Fig. 13. Sketch of the vortex breaking effects of the helical strakes during half a forced motion period under a small KC number for VR ¼ 4.
4.2.3. St numbers of the straked pipes From the aforementioned investigation, it can be seen that the two different effects of helical strakes have a significant influence on the time-frequency distribution of the VIV response. However, the afore mentioned analysis can only provide some qualitative conclusions, and hence, the influence of the competition between the vortex breakage and shear layer interaction prevention needs further study. To determine the influence of the helical strakes on the dominant frequency of the straked pipe under oscillatory flow, the general frequency spectrum proposed by (Wang et al., 2014) is introduced here and defined as,
Fig. 14. Sketch of the competition between effects of the helical strakes during half a forced motion period under a small KC number for VR ¼ 8.
number with different reduced velocities is shown in Figs. 13 and 14. The maximum velocity of the forced motion under a low reduced ve locity (VR ¼ 4) is lower, while the period of the forced motion is longer than that under a higher reduced velocity (VR ¼ 8). The helical strakes always or at most times play the role of a breaking vortex under a lower reduced velocity. Thus, the wake pattern contains many small vortex pairs, which leads to a higher VIV response frequency and “mode tran sition” phenomenon. As the maximum velocity of the forced motion increases and the forced motion period decreases, the effects of the helical strakes in preventing the interaction between the two shear layers become dominant under a higher reduced velocity as shown in Fig. 14. For example, suppose the forced motion maximum velocity at VR ¼ 4 is the critical velocity between the two different effects presented in Fig. 13. Below this critical velocity, such as at time (I) and (III), the vortex-breaking effects play a leading role, while above this critical velocity, the shear layer interaction-preventing effects dominate. As shown in Fig. 14, the time of the vortex-breaking effects is very short, whereas the prevention of shear layer effect is long. The prevention of shear layer reduces the dominant response frequency and the higher harmonic response frequency. More insightful experiments and CFD simulations will be conducted in the near future to further show this interesting phenomenon.
FðωÞ ¼
10 X
bf i ðωÞ
(21)
i¼1
where bf i ðωÞ is the FFT result at the ith gauge points and it is a function of the strain amplitude with respect to the response frequency. FðωÞ is the general amplitude-frequency spectrum by summing the strain amplitude at the same frequency component for all of the 10 gauge points. The general frequency spectrum of the straked pipe in the case of VR ¼ 8 and KC ¼ 56 is shown in Fig. 16. There is an obvious frequency peak at 1.75 Hz (7fo), the dominant frequency in this case. Contributions from 0.76 Hz (3fo), 1.0 Hz (4fo), 1.27 Hz (5fo), 1.52 Hz (6fo), 2.01 Hz (8fo) and 2.27 Hz (9fo) can also be seen but are quite insignificant. Typically, the VIV dominant frequency is governed by the St number. To further investigate the exact value of the St number, the relationship among the dominant frequency of the straked pipe, number of vortex shedding pairs, KC number under oscillatory flow, and Strouhal number can be defined as follows: N¼
fdomi fo
N fdomi Dfo fdomi D ¼ ⋅ ¼ ¼ St KC Um fo Um N ¼ St⋅KC 8
(22)
(23)
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Fig. 15. VIV response and time-frequency distribution at the 5th gauge point of the flexible pipe fitted with the helical strakes under a large KC number: (a) VR ¼ 4; (b) VR ¼ 6; (c) VR ¼ 8; (d) VR ¼ 12.
where fdomi is the dominant frequency; fo and Um are the forced oscil lation frequency and velocity amplitude, respectively. It should be noted that the diameters D in Eqs. (22) and (23) for both bare and straked pipe cases are all adopted the same value of bare pipe. The aforementioned relationship was firstly introduced to investigate the St number of a smooth rigid cylinder under oscillatory flow by (Sumer and Fredsøe, 1988). It has also been used by (Wang et al., 2015, 2016, 2017b)in the SCR and free-hanging riser model tests. By summarizing the dominant frequency for all of the test cases, the variation in the vortex shedding pairs with the KC number is shown in Fig. 17. The St number can be calculated based on Eqs. (22) and (23). Significant differences between the bare and straked pipe under oscil latory flow are observed. As shown in Fig. 17 (a), the vortex shedding pairs maintain a good linear relationship with KC numbers under VR ¼ 4, which is consistent with the results reported by (Sumer and Fredsøe, 1988). The St number is equal to 0.439 and 0.26 for the straked and bare pipes, respectively. It is unexpected that the St number of the straked pipe would be dramatically larger than that of a bare pipe, which can be attributed to the vortex-breaking effects of the helical strakes as previ ously mentioned. This finding is contrary to that under uniform flow where the helical strakes can reduce the VIV dominant frequency as reported by (Gao et al., 2014, 2015, 2016). This means that the helical strakes can increase the VIV dominant frequency under a low reduced velocity under oscillatory flow, which may lead to serious fatigue
Fig. 16. General strain amplitude-frequency spectrum for the straked pipe in the case of VR ¼ 8 and KC ¼ 56 (A ¼ 0.26 m and T ¼ 3.97 s).
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Fig. 17. Summarized relationship between the vortex-shedding pairs and KC numbers ( represents shedding pairs of the bare pipe; represents shedding pairs of is the fitting curve of the bare pipe; is the fitting curve of the straked pipe; and is the Strouhal law line). the straked pipe;
damage. When the reduced velocity increases, competition between the ef fects of the helical strakes in vortex breaking and preventing the inter action between two shear layers appears. This competition results in two distinct branches of the St numbers for the straked pipe under oscillatory flow. The main phenomena are as follows:
the first branch and increase it in the second branch. The turning point KCT is at approximately 70. (2) Similar features are also observed when the reduced velocity is equal to 8, as shown in Fig. 17 (c). The St number of the first and second branch is equal to 0.124 and 0.22, respectively. The turning point KCT is at approximately 80. (3) As the reduced velocity reaches 12, two distinct branches still exist for the straked pipe. In the first and second branches, which are divided by a KCT � 100, the St number is equal to 0.107 and 0.148, respectively. However, the St numbers of the straked pipe in these two branches are both lower than that of the bare pipe. (4) When the KC number further increases and is larger than the upper limit of that in the second branch, vortex-shedding pairs N decrease and do not maintain a linear relationship with KC number. N decreases from 40, 24, 18 at KC ¼ 140, 113, 121 to 4 at KC ¼ 165 under VR ¼ 6, 8, 12, respectively.
(1) Fig. 17 (b) shows the St number relationship when VR is 6. For the bare pipe, the St number is always equal to 0.18, which is slightly lower than the typical value of 0.2. For a straked pipe, two branches with St ¼ 0.155 (first branch) and St ¼ 0.29 (second branch) appear. The St number of the straked pipe is smaller than that of the bare pipe in the first branch with the KC number ranging from 20 to 70, while it is larger in the second branch with the KC number ranging from 70 to 140. This finding indicates that the helical strakes will decrease the VIV dominant frequency in
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The St numbers of the straked pipe in the two branches and bare pipe are summarized in Table 4. For the straked pipe, the St number de creases as the reduced velocity increases, while the turning point KCT shows an opposite tendency. This finding indicates the effect of the helical strakes in preventing the interaction between the shear layers is stronger under a higher reduced velocity due to its dependency on the Reynolds number. To summarize, the competition between the two ef fects of the helical strakes results in very complicated VIV dominant frequency for the pipe under oscillatory flow. It is possible that the he lical strakes can cause more serious fatigue damage under some condi tions, and hence, more attention should be paid to it during the engineering design process. However, under real ocean conditions, higher mode number of the flexible riser will be excited and the riser will suffer a high Re number. In our experiments, only the 1st mode was excited and it is impossible to systematically investigate the Re number effects. More experiments and numerical simulation of higher mode number and Re number effects should be conducted in the near future.
interaction between the shed vortices of the two half cycles also occurs for the straked pipe but weaker than that for the bare pipe at small KC number. Thus, lower suppression efficiency occurs at a small KC num ber. As shown in Fig. 18 (b), (c) and (d), the minimum values of the suppression efficiencies are 62.6%, 50.8% and 64.3% in the small KC number case and the maximum values are 93.5%, 91.9% and 82.4% in the large KC number case for VR ¼ 6, 8, 12, respectively. From above results, the suppression efficiencies of straked pipe at large KC number seem to be similar to that in the uniform flow. The breaking and messy vortex cannot excite a significant vibration. This may be the main rea sons for the higher suppression efficiency at larger KC number. Overall, the suppression efficiency of the helical strakes with a pitch equal to 15 D and a height equal to 0.25 D can reach 50% at a minimum. This also indicates that the suppression efficiency of other helical strake types should be verified under oscillatory flow. 4.3.2. Reduction ratio of fatigue damage Although effective suppression of the VIV of helical strakes studied in this paper can be seen from the amplitude reduction ratio, the effects of the helical strakes on fatigue damage need further study by combining them with the complicated frequency characteristics of the helical strakes under oscillatory flow. The VIV displacement amplitude reduc tion ratio cannot be considered as the unique criteria to evaluate the suppression efficiency (Resvanis et al., 2016). In our experiment, a constant pretension of 500N was applied to the ends of the model to achieve the designed natural frequencies. However, the tension force T(t) varied with time due to both varying drag force from forced motions and VIV response. Fig. 19 presents the time history of tension forced on the end of flexible pipe measured by the force transducers. The red dash line and solid line represent the tension force on straked and bare pipe, respectively. As shown in Fig. 19, compared with straked pipe, the tension force of bare pipe varies more obviously, especially for the case of larger KC number. This can be attributed to that bare pipe possess more significant VIV response than the straked pipe. The time-varying tension force can lead to a time-varying stress and contribute to the fatigue damage. Thus, the fatigue damage was derived from two parts: time-varying tension force and VIV force. To calculate the fatigue damage from measured strain signal in the model test, the bending stress and variable tension stress should be first obtained. The bending stress induced by VIV can be calculated by combing Eqs. (9) and (17), where the tension stress is obtained from the measured strain signal. As previously mentioned, the measured strain contains εT0, εT (z,t) and εVIV(z,t). According to Eqs. (7) and (8), the variable tension strain εT (z,t) can then be expressed by Eq. (25).
4.3. Suppression efficiency of helical strakes in oscillatory flow 4.3.1. Reduction ratio of VIV amplitude response The vortex-induced vibration suppression efficiency of the helical strakes, 15 D in pitch and 0.25 D in height, was tested in a rigid cylinder experiments under steady flow by Schaudt et al. (2008), obtaining at least 94% reduction. However, the suppression efficiency of a helical strake under oscillatory flow has not been studied. The efficiency η is usually defined as:
η¼
Y
Ys Y
(24)
� 100%
where Y and Ys are the standard deviation of the displacement for a bare and straked pipe, respectively. Fig. 18 shows the maximum displacement standard deviation of the VIV of bare and straked pipes. Decreasing trend of VIV response for bare pipe as KC number increases can be found in each test groups. This trend is consistent with the results found by (Wang et al., 2015). This is because the pipe encounters the previously shed vortexes as it reverses its directional motion. Such interaction is much stronger under small KC numbers, leading to a larger amplitude. This phenomenon can also be observed for a straked pipe under oscillatory flow for the same reason. Comparing the VIV response between the bare and straked pipes, the helical strake type can still effectively suppress the VIV under oscillatory flow as expected. However, the suppression efficiency is not as high as that under uniform flow as reported by (Schaudt et al., 2008). Under a lower reduced velocity, the broken shedding vortexes of the straked pipe are quite weak and cannot excite a significant VIV. Under this condition, the VIV displacements are stable with the KC number as shown in Fig. 16 (a). While the interaction between the shed vortices of the two half cycles for the bare pipe is stronger, the VIV response decreases as the KC number increases. Therefore, the suppression efficiency can reach 92.7% for a small KC number, but decreases to 71.9% for a large KC number. However, for a higher reduced velocity (VR ¼ 6, 8 and 12), the
εT ðz; tÞ ¼ f½εCF a ðz; tÞ
Straked pipe First branch
4 6 8 12
Bare pipe KCT
St
KC
0.439 0.155 0.124 0.107
– 20–70 20–80 20–100
– 70 80 100
Second branch St
KC
0.439 0.29 0.22 0.148
– 70–140 80–120 100–120
εT0 �g=2
(25)
In the Eq. (25), the initial tension strain εT0 is obtained through averaging the initial strain as shown in Fig. 20. Combing Eqs. (25) and (17), the variable tension stress can be calculated. Fig. 21 presents the distributions of root mean square of stress caused by VIV and varying tension force under KC ¼ 30 and 121 for VR ¼ 6.0. The solid line and dash line represent stress of bare and straked pipe, respectively. The blue line and red line represent the case of KC ¼ 30 and 121, respectively. As shown in Fig. 21 (a), the more obvious VIV response of bare pipe pos sesses more significant VIV stress σVIV_RMS than the straked one. σVIV_RMS at the middle points is obviously larger than those close to the end of the model, while the tension stress σT_RMS distributes evenly along the pipe as shown in Fig. 21 (b). Compared with σVIV_RMS, the stress derived from variable tension force is relatively smaller and there are no obvious differences under different KC numbers. After obtaining the stress, the stress cycles can be identified from the time history by the rainflow procedure (Amzallag et al., 1994). Then, according to Eq. (20), fatigue damage derived from VIV and varying tension force at each gauge point can be calculated based on Eqs. (18–20), respectively. Fig. 22 shows the fatigue damage distributions along the flexible pipe under KC ¼ 30 and 121 for VR ¼ 6. As shown in
Table 4 Strouhal number and KC number region for the straked and bare pipes. VR
εT0 � þ ½εCF c ðz; tÞ
St 0.26 0.18 0.146 0.174
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Fig. 18. Maximum displacement standard deviation of the VIV for the bare and straked pipes ( is the displacement response of the bare pipe; response of the straked pipe).
Fig. 22 (a), it can be seen that the fatigue damage caused by VIV bending strain Dot_VIV at the middle location is larger due to more obvious VIV response at this location. Dot_VIV of bare pipe is significant larger than that of straked one. It indicates that the helical strakes can reduce the Dot_VIV and prolong service life of the riser in an oscillatory flow. Fig. 22 (b) shows the distribution of fatigue damage derived from the varying tension force Dot_T. Compared with Dot_VIV, the Dot_T distributes more evenly along the flexible pipe and very smaller. Dot_T is about 104 orders of magnitude smaller than Dot_VIV. It means fatigue damage from vari able tension force can be ignored in our experiments. Beyond that, the variable tension force is not only caused by the VIV, it also affected by varying drag force from forced motions. To investigate the effects of helical strakes on reducing fatigue damage caused by pure VIV response, the maximum fatigue damage of Dot_VIV in each case is summarized. Fig. 23 shows the variation in the maximum fatigue damage of Dot_VIV with KC number for a bare and straked pipe. The maximum fa tigue damage decreases as the KC number increases for the bare pipe. The reason is the same as discussion of the VIV response amplitude distribution shown in Fig. 18.
is the displacement
Compared to the fatigue damage of the bare pipe, we surprisingly found that the maximum fatigue damage of the straked pipe under a low reduced velocity VR ¼ 4 is close to that of the bare pipe at a large KC number. Combining the Strouhal relationship shown in Fig. 17 and response amplitude distribution shown in Fig. 18, a higher VIV domi nant frequency caused by the vortex-breaking effect of the helical strakes may be the main reason for this phenomenon. As shown in Fig. 23 (a), the fatigue damage reduction rate is over 95%, while it de creases to 41% at a KC ¼ 121. When the reduced velocity further increases to 6, 8 and 12 as shown in Fig. 23 (b), (c), and (d), respectively, a similar trend of VIV response amplitude distribution is also observed in the maximum fatigue damage. The fatigue damage of the straked pipe under a small KC number is not as ideal as that under a large KC number. Although the dominant fre quency of the straked pipe at the second branch as previously mentioned is higher than that of the bare pipe, the response amplitude is effectively reduced at this branch. Thus, the effects of the two branches on the Strouhal number caused by covering the helical strakes cannot be seen in the maximum fatigue damage distribution. Due to limitation of the 12
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Fig. 19. Time history of tension force on the ends of the straked and bare pipe under KC ¼ 30 and KC ¼ 121 for VR ¼ 6.0.
Fig. 20. Time history of εCF_a (z,t) and εCF_c (z,t) of the straked pipe under VR ¼ 6.0 and KC ¼ 30.
Fig. 22. Fatigue damage distributions of the straked and bare pipe caused by VIV and varying tension force under KC ¼ 30 and KC ¼ 121 for VR ¼ 6.0.
5. Conclusions In this paper, an experimental study of a flexible pipe fitted with helical strakes was conducted in oscillatory flow under a KC number ranging from 20 to 160 and a maximum reduced velocity varying from 4 to 12. Compared with bare pipe, the effects of the helical strakes on the VIV response, St number, suppression efficiency and fatigue damage under oscillatory flow were investigated. The main conclusions are as follows:
Fig. 21. Stress distributions of the straked and bare pipe caused by VIV and varying tension force under KC ¼ 30 and KC ¼ 121 for VR ¼ 6.0.
(1) Under a lower reduced velocity, the “mode transition” phenom enon for the straked pipe is quite obvious, and the helical strakes can significantly increase the VIV dominant frequency. The Strouhal number of the straked pipe can reach 0.439, which is nearly twice that of the typical St number measured during a steady flow experiment. (2) Under a higher reduced velocity, two distinct branches in the variation of the St number against the KC number are observed. The helical strake reduces the dominant frequency in the first
length of the flexible pipe, higher modes were not excited in our ex periments. In the future, more work to excite higher mode of a flexible pipe in an oscillatory flow will be conducted. Overall, the fatigue damage for the straked pipe under oscillatory flow under a low reduced velocity and small KC number is larger and should be paid more atten tion in practical engineering projects.
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Fig. 23. Maximum fatigue damage distribution with the KC number of the bare and straked pipe ( is the fatigue damage of the bare pipe and of the straked pipe).
branch with a small KC number, while it increases it in the second branch with a large KC number. (3) The helical strake shows its effects in suppressing VIV under oscillatory flow. However, the suppression efficiency of the VIV and fatigue damage, particularly under the conditions of a small KC number and low reduced velocity, is not as high as that under steady flow. More attentions should be paid to this phenomenon in practical engineering projects.
is the fatigue damage
Acknowledgments The authors gratefully acknowledge the support from the National Science Fund for Distinguished Young Scholars of China (No. 51825903), Shanghai Science and Technology Program (No. 19XD1402000), the National Natural Science Foundation of China (No. 51490674), National Program on Key Basic Research Project of China 14
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(973 Program) (No. 2015CB251203) and Ub-project of the Important National Science & Technology Specific Projects of China (No. 2016ZX05028-002-004). And all authors also would like to acknowl edge permission and support from SBM Offshore to prepare and publish this work.
Jeon, S., Choi, J., Jeon, W., Choi, H., Park, J., 2004. Active Control of Flow over a Sphere for Drag Reduction at a Subcritical Reynolds Number. Cambridge University Press. Pesce, C.P., Franzini, G.R., Fujarra, A.L.C., Gonçalves, R.T., Salles, R., Mendes, P., 2017. Further Experimental Investigations on Vortex Self-Induced Vibrations (VSIV) with a Small-Scale Catenary Riser Model. V002T008A016. Pinto, A., Broglia, R., Di Mascio, A., Campana, E.F., Rocco, P., 2006. Numerical investigation of the unsteady flow at high Reynolds number over a marine riser with helical strakes. In: Proceedings of OMAE2006 25th International Conference on Offshore Mechanics and Arctic Engineering, pp. 587–595. Resvanis, T.L., Rao, Z., Kim Vandiver, J., 2016. Effects of strake coverage and marine growth on flexible cylinder vortex-induced Vibrations 1. J. Offshore Mech. Arct. Eng. 138, 051801-051801-051811. Schaudt, K.J., Wajnikonis, C., Spencer, D., Xu, J., Leverette, S., Masters, R., 2008. Benchmarking of VIV suppression systems. In: ASME 2008 International Conference on Offshore Mechanics and Arctic Engineering, pp. 33–42. Sumer, B.M., Fredsøe, J., 1988. Transverse vibrations of an elastically mounted cylinder exposed to an oscillating flow. J. Offshore Mech. Arct. Eng. 110, 387–394. Thiagarajan, K.P., Constantinides, Y., 2005. CFD Analysis of Vortex-Induced Motions of Bare and Straked Cylinders in Currents. American Society of Mechanical Engineers, pp. 903–908. Trim, A.D., Braaten, H., Lie, H., Tognarelli, M.A., 2005. Experimental investigation of vortex-induced vibration of long marine risers. J. Fluids Struct. 21, 335–361. Vandiver, J.K., Swithenbank, S.B., Jaiswal, V., Jhingran, V., 2006. Fatigue damage from high mode number vortex-induced vibration. In: International Conference on Offshore Mechanics and Arctic Engineering, pp. 803–811. Wang, J., Fu, S., Baarholm, R., Wu, J., Larsen, C.M., 2014. Fatigue damage of a steel catenary riser from vortex-induced vibration caused by vessel motions. Mar. Struct. 39, 131–156. Wang, J., Fu, S., Baarholm, R., Wu, J., Larsen, C.M., 2015. Out-of-plane vortex-induced vibration of a steel catenary riser caused by vessel motions. Ocean. Eng. 109, 389–400. Wang, J., Fu, S., Larsen, C.M., Baarholm, R., Wu, J., Lie, H., 2017. Dominant parameters for vortex-induced vibration of a steel catenary riser under vessel motion. Ocean. Eng. 136, 260–271. Wang, J., Fu, S., Wang, J., Li, H., Ong, M.C., 2017. Experimental investigation on vortexinduced vibration of a free-hanging riser under vessel motion and uniform current. J. Offshore Mech. Arct. Eng. 139, 041703-041703-041718. Wang, J., Xiang, S., Fu, S., Cao, P., Yang, J., He, J., 2016. Experimental investigation on the dynamic responses of a free-hanging water intake riser under vessel motion. Mar. Struct. 50, 1–19. Williams, J.E.F., Zhao, B.C., 1989. The active control of vortex shedding. J. Fluids Struct. 3, 115–122.
References Allen, D., Henning, D., Li, L., 2004. Performance Comparisons of Helical Strakes for VIV Suppression of Risers and Tendons. Amzallag, C., Gerey, J.P., Robert, J.L., Bahuaud, J., 1994. Standardization of the rainflow counting method for fatigue analysis. Int. J. Fatigue 16, 287–293. Bearman, P.W., Owen, J.C., 1998. Reduction of bluff-body drag and supperssion of vortex shedding by the introduction of wavy separation lines. J. Fluids Struct. 12, 123–130. Blevins, R.D., Saunders, H., 1977. Flow-induced Vibration. Van Nostrand Reinhold Co. Cheng, J., Cao, P., Fu, S., Constantinides, Y., 2016. Experimental and numerical study of steel lazy wave riser response in extreme environment. In: ASME 2016 International Conference on Ocean, Offshore and Arctic Engineering. V005T004A055. Constantinides, Y., Cao, P., Cheng, J., Fu, S., Kusinski, G., 2016. Steel lazy wave riser tests in harsh offshore environment. In: ASME 2016 International Conference on Ocean, Offshore and Arctic Engineering. V005T004A057. Constantinides, Y., Oakley, J.O.H., 2006. Numerical Prediction of Bare and Straked Cylinder VIV, pp. 745–753. Fu, S., Wang, J., Baarholm, R., Wu, J., Larsen, C.M., 2014. Features of vortex-induced vibration in oscillatory flow. J. Offshore Mech. Arct. Eng. 136, 011801. Galvao, R., Lee, E., Farrell, D., Hover, F., Triantafyllou, M., Kitney, N., Beynet, P., 2008. Flow control in flow–structure interaction. J. Fluids Struct. 24, 1216–1226. Gao, Y., Fu, S., Ma, L., Chen, Y., 2016. Experimental investigation of the response performance of VIV on a flexible riser with helical strakes. Ships Offshore Struct. 11, 113–128. Gao, Y., Fu, S., Ren, T., Xiong, Y., Song, L., 2015. VIV response of a long flexible riser fitted with strakes in uniform and linearly sheared currents. Appl. Ocean Res. 52, 102–114. Gao, Y., Fu, S., Song, L., Peng, T., Lei, R., 2014. Experimental Investigation on the Suppression Device of VIV of a Flexible Riser. V002T008A026. He, J.W., Glowinski, R., Metcalfe, R., Nordlander, A., Periaux, J., 2000. Active control and drag optimization for flow past a circular cylinder : I. Oscillatory cylinder rotation. J. Comput. Phys. 163, 83–117.
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Vortex-induced vibration of flexible pipe fitted with helical strakes in oscillatory flow Haojie Ren a, b, Yuwang Xu a, b, *, Jingyun Cheng c, Peimin Cao c, Mengmeng Zhang a, b, Shixiao Fu a, b, Ziyang Zhu a a b c
State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai, China Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration, Shanghai, China SBM Offshore, Houston, TX, USA
A R T I C L E I N F O
A B S T R A C T
Keywords: Vortex-induced vibration Helical strake Flexible pipe Oscillatory flow
Helical strakes are widely used to suppress vortex-induced vibration (VIV) in offshore engineering. However, the VIV response and suppression efficiency of the straked pipe remain unclear in an oscillatory flow. In this paper, an experimental study of VIV for a flexible pipe fitted with helical strakes was conducted in an oscillatory flow with the Keulegan-Carpenter (KC) number varying from 21 to 165 and maximum reduced velocities ranging from 4 to 12. The effects of the helical strakes on the VIV response, Strouhal number, suppression efficiency and fatigue damage in an oscillatory flow were investigated. The results show that the suppression efficiency and fatigue damage reduction ratio are not as ideal in oscillatory flow as those in steady flow. Moreover, the VIV dominant frequency is obviously magnified and a large Strouhal number is observed, reaching 0.439 under a lower reduced velocity of 4. As the reduced velocity further increases to 6, 8, 12, two branches of Strouhal numbers are clearly seen. The dominant frequency decreases at the first branch with a small KC number and increases at the second branch with a large KC number.
1. Introduction As oil and gas production moves into deeper waters, risers are becoming increasingly slender. Under the action of ocean currents, vortices are generated and alternately shed from the sides of these flexible slender risers, resulting in a periodic excitation force. If the frequency of the vortex-induced force is near the natural frequencies of the risers, a resonance vibration will occur. This is the so-called vortexinduced vibration (VIV) (Blevins and Saunders, 1977). This periodic vibration has been proven to be the main reason for fatigue damage of risers. Therefore, a means to effectively suppress VIV has been studied for decades in industry and academia. At present, there are two main methods to suppress VIV of the riser: active and passive control. The main difference between them is that active control requires additional power (He et al., 2000; Jeon et al., 2004; Williams and Zhao, 1989), while passive control does not. The latter is more widely used due to its easier manufacture. Different pas sive control methods to suppress VIV have been studied by many re searchers (Allen et al., 2004; Bearman and Owen, 1998; Galvao et al.,
2008; Gao et al., 2015; Trim et al., 2005). Helical strake is among the most widely used apparatuses in practical projects. Its high suppression efficiencies in the steady flow have been verified by both numerical simulations and experiment studies (Vandiver et al., 2006). tested three different distributions of strakes in the reduction of the VIV amplitude and frequency (Allen et al., 2004). presented the performance of different helical strake geometries at critical Reynolds numbers (Trim et al., 2005). studied the suppression effectiveness of the strake on a long flexible riser in both the CF and IL directions (Gao et al., 2015). pre sented the effects of strake configurations in both uniform and linearly sheared flows (Constantinides and Oakley, 2006; Pinto et al., 2006; Thiagarajan and Constantinides, 2005). studied the VIV suppression mechanism of helical strakes by using the computational fluid dynamic (CFD) method. They found that helical strake could prevent the inter action between two shear layers and break vortex shedding, thus sup pressing the VIV. The aforementioned studies focused on the performance of helical strakes under steady flow, and there are extremely few publications regarding the performance of helical strakes under oscillatory flow.
* Corresponding author. State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai, China. E-mail address: [email protected] (Y. Xu). https://doi.org/10.1016/j.oceaneng.2019.106274 Received 10 May 2019; Received in revised form 13 July 2019; Accepted 5 August 2019 Available online 26 August 2019 0029-8018/© 2019 Elsevier Ltd. All rights reserved.
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However, the marine risers, for instance, the steel catenary riser (SCR) and steel lazy wave riser (SLWR) act as the only channel to con nect the subsea wellhead and the platform. In a real sea state, the waveinduced periodic motion of the platform always results in a relatively equivalent oscillatory flow around the risers. It has been found from recent experimental study that the oscillatory flow induced by the pure motion of the platform can also excite the VIV at the sag-bend of the SCR (Wang et al., 2015, 2017a), termed vessel motion-induced VIV (VMI- VIV) (Pesce et al., 2017). The VMI-VIV can cause serious fatigue damage to risers (Wang et al., 2014). To improve the understanding of this phenomenon (Fu et al., 2014), conducted a flexible pipe model test in oscillatory flow by forcing the model to oscillate in still water with different periods and amplitudes. A VIV developing process in oscilla tory flow was first proposed, including building up, lock-in and dying out. This novel VIV phenomenon has attracted much attention from both industry and academia. Another experiment with a SLWR was presented by (Cheng et al., 2016) and (Constantinides et al., 2016). The similar VMI-VIV phenomenon was also observed. To suppress the VIV induced by vessel motion, tests of a pipe model fitted with helical strakes with a pitch of 15 D and a height of 0.25 D were further conducted. However, VIV still occurred for the straked pipe, which is contrary to the popular belief that helical strakes can nearly fully suppress VIV. What causes this result is still unclear. To further improve the understanding of VIV performance for straked pipes in the equivalent oscillatory flow, we conducted VIV model tests for a straked pipe under oscillatory flow in SKLOE. A straked pipe with pitch of 15 D and height of 0.25 D was forced to harmonically oscillate under various combinations of amplitude and period with Keulegan-Carpenter (KC) numbers varying between 21 and 165 and maximum reduced velocities ranging from 4 to 12. The effects of the helical strakes on VIV response, Strouhal number, suppression efficiency and fatigue damage are discussed.
Fig. 2. Simplified sketch of the setup.
2. Model test
Fig. 3. Detailed view of the end connection.
2.1. Test apparatus
random (PPR) pipe outside and a copper cable inside to satisfy the designed mass and bending stiffness. Silicone gel was placed between different layers to avoid relative slippage. As shown in Fig. 4, the height, pitch and number of starts of the helical strake were three of its main geometric parameters, which were 0.25D, 15D and 3 in the experi mental study, respectively. The helical strakes were tied to the flexible pipe. The main properties of the test model are listed in Table 1. The 1st and 2nd order eigen frequency of the flexible pipe in still water can be calculated by Eq. (1) and were 1.77 Hz and 3.86 Hz, respectively.
A model test was conducted in an ocean basin at Shanghai Jiao Tong University. The entire experimental setup, primarily containing two horizontal and vertical tracks and a flexible pipe model, was installed under the bottom of a sub-carriage as shown in Fig. 1 and Fig. 2. Details of the connection frame at the end side of the experimental apparatus are shown in Fig. 3. The pipe model was connected to a force sensor through a universal joint. The force sensor was connected to a tensioner that was fixed to the slider of the vertical track. A pretension force of 500N was applied to the flexible pipe. Meanwhile, two circular end plates were installed to avoid the perturbation caused by the motion of the supporting frame. The flexible pipe model was constructed with a polypropylene
Fig. 1. Overview of the whole experimental setup.
Fig. 4. Geometric parameters of the helical strakes. 2
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natural mode is the potential excitation mode for all the test cases. Notably, the added mass may deviate from the value of 1.0 in still water, and the forced oscillation periods will change with the KC number at the same maximum reduced velocity. All the test cases were divided into four groups with different maximum reduced velocities, as presented in Table 2. The correspond ing maximum Reynolds number (Re) has been also list in this table, which can be calculated by Eq. (6). Each test group had test cases with a KC number ranging from 21 to 165 under the same amplitude of the forced motion velocity. The same cases were also conducted for a bare pipe.
Table 1 Parameters of the flexible pipe model. Item
Value
Model length L (m) Outer diameter D (mm) Mass in air m (kg/m)
4 29 1.696
Bending stiffness EI (N⋅m2) Tensile stiffness EA (N) Pre-tension T (N) Damping ratio ζ (%) Calculated first natural frequency f10 in water (Hz) Calculated second natural frequency f20 in water (Hz)
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n T0 n2 π2 EI 1 fn0 ¼ þ 2 ⋅ ; m ¼ m þ πD2 ρCm ðCm ¼ 1:0Þ 2L m 4 L m
46.433 1.528 Eþ06 500 2.53 1.77 3.86
Remax ¼ n ¼ 1; 2::::
3. Basic theory 3.1. Preprocessing For convenience in description, the coordinate system is defined as O-XYZ: the origin is at one of the ending points of the riser and the Zaxis, X-axis and Y-axis are along the axis of the riser, IL direction and CF direction, respectively, as shown in Fig. 2. At the start of the experiment, a constant pre-tension of 500N was exerted at the flexible pipe ends. However, the pretension periodically varied with pipe oscillation. Thus, the measured strain included three components: the initial axial strain εT0 caused by pre-tension, the vari able axial strain εT(t) caused by varying tension and the bending strain εVIV caused by the vortex-induced vibration. The original measured strains εCF_a (z,t) and εCF_c (z,t) sampled by CF_a and CF_c measurement points, respectively can be correspondingly expressed as,
2.2. Test matrix To investigate the VIV response characteristics of the flexible pipe fitted with helical strakes in oscillatory flow, the model was forced to oscillate in harmonic motion under various combinations of amplitude and period in the horizontal direction. The instantaneous displacement X(t) and velocity of the forced motion U(t) can be expressed as follows: � � 2π XðtÞ ¼ Am sin t (2) T
VR ¼
εCF a ðz; tÞ ¼ εT0 þ εT ðz; tÞ þ εVIV ðz; tÞ
(7)
εCF c ðz; tÞ ¼ εT0 þ εT ðz; tÞ
(8)
(3)
εVIV ðz; tÞ ¼ ½εCF a ðz; tÞ
2πAm D
(4)
2π⋅Am T⋅f10 ⋅D
(5)
εVIV ðz; tÞ
Combing Eqs. (7) and (8), the pure VIV strain at z location εVIV (z,t) can be calculated by
where Am and T are oscillation amplitude and period, respectively. The KC number and maximum reduced velocity VR are considered two of the main parameters that determine the VIV characteristics of a flexible cylinder under oscillatory flow and are represented as: KC ¼
(6)
υ
where Um is the forced motion velocity amplitude. υ is the kinematic viscosity coefficient. In our experiment, the ambient temperature is maintained near 15 � C and υ is therefore approximately 1.14 � 10 6 m2 s 1.
(1)
where T0 is the pretension force. m and m are the mass per unit length in still water and air, respectively. L and D are the length and diameter of flexible pipe, respectively. EI is the bending stiffness of pipe. ρ is the density of the water, ρ ¼ 1000 kg/m3. The added mass coefficient Cm is assumed to be 1.0. It should be noted that the diameter values are all kept as that of bare pipe. Four groups of Fiber Bragg Grating (FBG) strain sensors were installed to measure the strain responses in both the CF and IL directions. Each of the FBG groups (CF_a, CF_c, IL_b and IL_d) had ten measurement points along the pipe separated by 0.36 m, as shown in the schematic diagram in Fig. 5.
� � 2π 2π UðtÞ ¼ Am cos t T T
Um D
(9)
εCF c ðz; tÞ�=2
Fig. 6 shows the original VIV strain time history of VR ¼ 6 and KC ¼ 143. The original VIV strain obviously contains significant fo and Table 2 Test matrix.
where f10 denotes the first natural frequency of the test model in still water. f10 is usually used to calculate the reduced velocity since the first
Case No.
VR
Am(m)
KC
Um (m/s)
Remax
1–19 20–40 41–61 62–82
4 6 8 12
[0.10:0.02:0.76] [0.10:0.02:0.76] [0.10:0.02:0.76] [0.10:0.02:0.76]
21–165 21–165 21–165 21–165
0.206 0.309 0.411 0.617
5240 7860 10455 15696
Fig. 5. FBG strain sensor instrumentation along the flexible pipe model. 3
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Fig. 6. Time history of the original VIV strain at VR ¼ 6 and KC ¼ 143.
2fo frequency components, which is caused by the swing of the flexible pipe. fo refers to the forced oscillation frequency and is equal to 1/T. As Fig. 7 (a) shows, the flexible pipe has an initial deflection under the action of gravity. Combined with the forced motion, the flexible pipe with the initial deflection performs a pendulum movement. To further verify this view, Fig. 7(b) shows the trajectory of the strain at the 5th gauge points. The blue solid line represents the original total strain including the VIV strain and bending strain caused by the pendulum movement. The red solid line is these bending strain components below 2.5 fo after low-pass filtering. Through Fig. 7(b), an obvious asymmet rical pendulum trajectory presented by red solid line can be observed. It also explains the significant f0 and 2f0 components that exist in the original VIV strains. This asymmetrical pendulum movement may be caused by the asymmetrical drag force in an oscillation period. To eliminate the effects of the flexible pipe pendulum motion caused by the forced motion and high frequency noise, a bandpass filter was utilized. The passband cutoff frequencies were set as 2.5fo and 30 Hz. A comparison of the original VIV strain and filtered VIV strain at the 5th gauge points (VR ¼ 6 and KC ¼ 143) is shown in Fig. 8. Although the strain caused by the cycloidal motion plays an important role in bending strain response, this strain cannot be counted in the VIV strain because this strain is not caused by the vortex shedding.
n X
pi ðtÞϕi ðzÞ;
yðz; tÞ ¼
(10)
z 2 ½0; L�
i¼1
where y(z,t) is the VIV displacement response in the CF direction at the z location, and pi(t) is the ith generalized coordinate displacement values at time t; ϕi(z) is ith displacement modal shape at the z location. Based on the small deformation assumption, the curvature κ(z,t) can be expressed as, κðz; tÞ ¼
n ∂2 yðz; tÞ X ¼ pi ðtÞϕ00 i ðzÞ; 2 ∂z i¼1
z 2 ½0; L�
(11)
where ϕ’’i is the ith modal shape of the curvature. According to the geometric relationship between the curvature and strain, the strain can be calculated by,
εðz; tÞ ¼ κðz; tÞR ¼ R
n X
pi ðtÞϕ00 i ðzÞ;
z 2 ½0; L�
(12)
i¼1
where R denotes the radius of the flexible pipe model at the z location. In our model, mode shapes of the displacement are sinusoidal, which can be expressed as,
3.2. Displacement identified method
ϕi ðzÞ ¼ sin
According to the Euler-Bernoulli beam theory, the VIV displacement response of the flexible pipe under the external load can be expressed as the sum of the mode shapes multiplied by the generalized coordinate values of each step. The VIV displacement response in the CF direction can be expressed as,
iπz ; L
i ¼ 1; 2:::
(13)
From Eq. (11), the mode-shapes of the curvature are also sinusoidal. After obtaining the mode shape of the displacement and curvature, the modal generalized coordinates can be obtained from Eq. (12). Then, the VIV displacement response in the CF direction can be calculated by Eq. (10). Fig. 9 presents the distribution of root mean square of VIV
Fig. 7. Experimental image and trajectory of strain at the 5th gauge points (VR ¼ 6 and KC ¼ 143). 4
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Fig. 8. Comparison of the original VIV strain and filtered strain in the CF direction at the 5th gauge points (VR ¼ 6 and KC ¼ 143).
Fig. 9. Distribution of root mean square of VIV displacement and VIV strain along the flexible pipe in the case of VR ¼ 6 and KC ¼ 30 (A ¼ 0.14m, T ¼ 2.86s): (a) Root mean square of VIV displacement; (b) Root mean square of VIV strain.
displacement (YRMS/D) and VIV strain (εRMS)along the flexible pipe in the case of VR ¼ 6 and KC ¼ 30. As shown in Fig. 9 (a), the blue and red lines present the VIV displacement of bare and straked pipe, respec tively. The VIV displacement of bare pipe is larger than that of straked pipe. It indicates that the helical strakes can also suppress the vortexinduced vibration in an oscillatory flow. To validate the displacement identification, the identified VIV strain was recalculated based on the reconstructed VIV displacement. Through Eq. (11), curvature can be obtained by the second order difference of VIV displacement to position z. Then identified strain values are further calculated by Eq. (12). As presented in Fig. 9 (b), the red dash line and blue solid line are the identified εRMS of straked and bare pipe, respec tively. The red circle and blue square symbols represents the measured εRMS of straked and bare pipe, respectively. The identified values are in a good agreement with the measured one. This consistency demonstrates the validity of displacement identification method for the flexible pipe in an oscillatory flow.
shedding frequency, the VIV characteristics in the CF direction vary with time. This “time-varying” feature has been reported to be the major difference between the VIV in steady flow and that in oscillatory flow (Fu et al., 2014). To investigate the time-varying feature of the flexible pipe fitted with helical strakes, the wavelet transformation is introduced to help analyze the time-frequency distribution of the VIV in oscillatory flow. The continuous wavelet transform equation is expressed as, � � � Z þ∞ t τ f ðtÞψ � ð dt (15) WTf a; τ ¼ ⟨ f ðtÞ; ψ a;τ ðtÞ⟩ ¼ a 1=2 a ∞ where WTf(a, τ) is the coefficient of the time domain signal after the wavelet transformation, which represents the frequency variation at that time scale. Parameter a is the scale factor, τ is the shift factor, and ψ (t) is the mother wavelet. In this paper, we use the Morlet wavelet equation as the mother wavelet, which can be defined as, �� 2 ψ t ¼ Ce t =2 cosð5tÞ (16)
3.3. Time-frequency analysis The shedding frequency changes with the periodic oscillation ve locity of the model, which can be written as: fst ðtÞ ¼
St⋅jUðtÞj D
3.4. Fatigue damage estimation The VIV can cause serious fatigue damage to marine risers. There fore, it is necessary to estimate the fatigue damage of the flexible pipe fitted with helical strakes under different maximum reduced velocities VR and KC numbers. The stresses σ at the outer pipe can be calculated by,
(14)
where St refers to the Strouhal number (typically, St ¼ 0.2); fst is the vortex shedding frequency. Under the effect of the periodically varying 5
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(17)
σ ðz; tÞ ¼ E⋅εðz; tÞ
where E is the elastic modulus of the flexible pipe. According to the Miner-Palmgren hypothesis, fatigue damage occurs when the accumulation of fatigue damage from the stress time histories with varying amplitudes satisfies Eq. (18) X ni Da ¼ ¼1 (18) Ni i where ni is the number of cycles with the stress range Δσ i; Ni is the number of cycles to failure for the specific stress range Δσi, which can be expressed as Eq. (19) according to the S–N curve. Ni ¼ 10ðloga
(19)
m logΔσi Þ
Thus, the total fatigue damage Dot can be calculated by the sum mation of the contribution from all cycles, Ntot X
Dot ¼
(20)
Dai i
where Ntot is the total number of the stress cycles in the VIV time history. The B1 S–N curve from (DNV-RP-C203 2011) is applied. The main parameters are listed in Table 3.
Fig. 10. Time history of the VIV response and time frequency distribution at the 5th gauge point of the bare pipe in the case of VR ¼ 6 and KC ¼ 30: (a) Timevarying frequency of the vortex shedding; (b) VIV displacement response; (c) Modal weight of displacement; (d) Time frequency distribution; (e) Timevarying natural frequency and dominant response frequency.
4. Results and discussions 4.1. VIV responses of the bare pipe in oscillatory flow To investigate the effects of helical strakes on VIV response in an oscillatory flow, an experimental study of a bare pipe was firstly con ducted as a benchmark. For the bare pipe test cases, the first mode is always dominant in our experiments. Therefore, the results measured at the 5th gauge point that is in the middle of the pipe model (z ¼ 1.73 m) is used for further analysis. Fig. 10 and Fig. 11 show the time history of the VIV response and time frequency distribution in the case of VR ¼ 6 under a small KC number (KC ¼ 30) and a large KC number (KC ¼ 121), respectively. Each figure contains five subplots. Figure (a) is the timevarying shedding frequency fst, which is calculated based on Eq. (14). Figure (b) is the VIV displacement response, which is reconstructed by utilizing the modal superposition method as proposed in Section 3.2. Figure (c) shows the time history of the modal weight p(t) of displace ment. Figure (d) is the wavelet analysis of the strain signal. The depth of the color indicates the concentration level of the strain components. Figure (e) presents the time varying natural frequency and VIV response dominant frequency. The blue dash and red solid lines represent the first and second order eigen frequency, while the black line refers to the dominant frequency. It should be noted that the first and second order eigen frequency (f1, f2) have been calculated based on the measured time-varying tension, assuming the added mass coefficient to be equal to 1. As shown in Fig. 10, the shedding frequency calculated by Eq. (14) varies from 0 to 2 Hz. The VIV displacement amplitude is less modulated and the maximum value reaches 0.64 D. As seen from the wavelet results shown in Fig. 10 (e), the dominant frequency is always close to f1. This indicates that the lock-in occurs at first mode. Although the response is dominated by the 1st natural mode, higher harmonics also occur under the small KC number (KC ¼ 30) presented in Fig. 10 (d). However, under the large KC number (KC ¼ 121) as shown in Fig. 11, a strong amplitude
Fig. 11. Time history of the VIV response and time frequency distribution at the 5th gauge point of the bare pipe in the case of VR ¼ 6 and KC ¼ 121: (a) Time-varying frequency of the vortex shedding; (b) VIV displacement response; (c) Modal weight of displacement; (d) Time frequency distribution; (e) Timevarying natural frequency and dominant response frequency.
modulation phenomenon appears and the VIV developing process is comprised of three phases: building up, lock-in and dying out. The maximum VIV displacement amplitude value reaches 0.65 D, which is the same as that in the small KC number case. These results are consis tent with the VIV features in oscillatory flow found by (Fu et al., 2014; Wang et al., 2014). Significant differences in the VIV response between the small and large KC numbers will provide a good reference for investigating the effects of helical strakes. The comparison of these distinct features between the bare and straked pipes will help in improving our understanding of the effects of helical strakes in an
Table 3 S–N curve parameters. S–N curve
loga
m
B1
14.917
4.0
6
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oscillatory flow.
Fig. 12(b) and(c). The response becomes very steady and is always dominated by the 1st natural mode, which is lower than the maximum shedding frequency. This finding indicates that the strakes can reduce the dominant frequency and suppress the higher frequency response components under higher reduced velocities, which is consistent with the results in uniform flow found by (Gao et al., 2016). The main reason for the difference between the low and high reduced velocities as previously described may be attributed to the competition between the effects of the helical strakes on the breaking vortex and the prevention of interaction between two shear layers. Breaking vortex effects means the helical strakes break the large orderly vortex into many weak and messy ones. The latter effect is that helical strakes prevent the two shear layers from affecting each other and make two layers parallel to the flow direction shown in Fig. 14. Under a lower flow velocity (a lower Re number), the vortex shedding from the two sides of the straked pipe is quite weak and very easily broken. The breaking vortex leads to a higher response frequency. However, the vortex be comes stronger and sheds from the two sides of the straked pipe in parallel under a higher flow velocity (a higher Re number). The inter action of the two shear layers is prevented by the helical strakes (Con stantinides and Oakley, 2006; Pinto et al., 2006). The response frequency is reduced under this condition. Only one effect works under steady flow, but two different effects on the response frequency will compete in oscillatory flow. For easier understanding, a sketch of the competition between the effects of the helical strakes under a small KC
4.2. VIV responses of the straked pipe in oscillatory flow 4.2.1. Time-varying responses of the straked pipe under a small KC number The effects of the helical strakes on the VIV response in an oscillatory flow with a small and large KC number were investigated by comparing them to the experimental results of the bare pipe model. Fig. 12 shows the VIV response and time frequency distribution at the 5th gauge points of the straked flexible pipe in the case of a small KC number (KC ¼ 38) with different reduced velocities ranging from 4 to 12. As shown in Fig. 12(a), an obvious “mode transition” phenomenon is observed under VR ¼ 4. The dominant frequency represented by the black solid line of the straked pipe changes back and forth between the 1st and 2nd natural frequencies. Assuming the St number is equal to 0.2, the maximum shedding frequency of the straked pipe based on Eq. (14) should be 1.42 Hz. However, this is much lower than the dominant response frequency found in the experiment, as shown in the third subplot in Fig. 12(a). This means that the helical strakes can result in a St number much larger than 0.2 (a typical value for a bare pipe) and significantly magnify the shedding frequency under a small reduced velocity. However, as the maximum reduced velocity increases to 6, 8 and 12, “mode transition” and higher harmonic components disappear, and the response characteristics are similar to those of the bare pipe, as shown in
Fig. 12. VIV response and time-frequency distribution at the 5th gauge point of the flexible pipe fitted with helical strakes at a small KC number: (a)VR ¼ 4; (b) VR ¼ 6; (c) VR ¼ 8; (d) VR ¼ 12. 7
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4.2.2. Time-varying responses of the straked pipe under a large KC number To further investigate the VIV responses under a large KC number, Fig. 15 shows the VIV response and time-frequency distribution at the 5th gauge point of the straked flexible pipe in the case of the large KC number, which is approximately 100. The reduced velocities vary from 4 to 12. As VR is equal to 4 shown in Fig. 15(a), a “mode transition” very often occurs, and the response frequency is quite unstable. A signifi cantly higher response frequency component can be observed. There are no obvious VIV processes under such a low reduced velocity, which is completely different from that of the bare pipe. The main reason is that the vortex-breaking effects of the helical strakes dominate. In the cases of a VR ¼ 6 and 8 shown in Fig. 15(b) and (c), respec tively, the time-frequency distribution becomes very stable. The response characteristics are similar to those of the bare pipes. A stronger amplitude modulation and obvious VIV developing process comprised of building up, lock-in and dying out can be seen. The vortex-breaking effects dominant under the low velocity cause higher harmonic re sponses in this case, differing from the bare pipe. As the reduced velocity increases to 12, a strong amplitude modu lation still occurs, while the VIV developing process and higher har monic response disappear. The shorter cycle of the forced motion and higher velocity may serve as the main reasons, which suppresses the vortex-breaking effects. In addition, the vortex-shedding process can be altered when the cylinder reverses its motion direction and encounters the previously shed vortices (Fu et al., 2014; Sumer and Fredsøe, 1988). The interaction stops the VIV from developing and leads to a stronger response as previously explained. The maximum values of the displacement at VR ¼ 6, 8 and 12 are 0.07D, 0.13D and 0.21D, respec tively. The displacement of VR ¼ 12 is obviously larger than that at VR ¼ 6 and 8. This is consistent with the previously explained reason.
Fig. 13. Sketch of the vortex breaking effects of the helical strakes during half a forced motion period under a small KC number for VR ¼ 4.
4.2.3. St numbers of the straked pipes From the aforementioned investigation, it can be seen that the two different effects of helical strakes have a significant influence on the time-frequency distribution of the VIV response. However, the afore mentioned analysis can only provide some qualitative conclusions, and hence, the influence of the competition between the vortex breakage and shear layer interaction prevention needs further study. To determine the influence of the helical strakes on the dominant frequency of the straked pipe under oscillatory flow, the general frequency spectrum proposed by (Wang et al., 2014) is introduced here and defined as,
Fig. 14. Sketch of the competition between effects of the helical strakes during half a forced motion period under a small KC number for VR ¼ 8.
number with different reduced velocities is shown in Figs. 13 and 14. The maximum velocity of the forced motion under a low reduced ve locity (VR ¼ 4) is lower, while the period of the forced motion is longer than that under a higher reduced velocity (VR ¼ 8). The helical strakes always or at most times play the role of a breaking vortex under a lower reduced velocity. Thus, the wake pattern contains many small vortex pairs, which leads to a higher VIV response frequency and “mode tran sition” phenomenon. As the maximum velocity of the forced motion increases and the forced motion period decreases, the effects of the helical strakes in preventing the interaction between the two shear layers become dominant under a higher reduced velocity as shown in Fig. 14. For example, suppose the forced motion maximum velocity at VR ¼ 4 is the critical velocity between the two different effects presented in Fig. 13. Below this critical velocity, such as at time (I) and (III), the vortex-breaking effects play a leading role, while above this critical velocity, the shear layer interaction-preventing effects dominate. As shown in Fig. 14, the time of the vortex-breaking effects is very short, whereas the prevention of shear layer effect is long. The prevention of shear layer reduces the dominant response frequency and the higher harmonic response frequency. More insightful experiments and CFD simulations will be conducted in the near future to further show this interesting phenomenon.
FðωÞ ¼
10 X
bf i ðωÞ
(21)
i¼1
where bf i ðωÞ is the FFT result at the ith gauge points and it is a function of the strain amplitude with respect to the response frequency. FðωÞ is the general amplitude-frequency spectrum by summing the strain amplitude at the same frequency component for all of the 10 gauge points. The general frequency spectrum of the straked pipe in the case of VR ¼ 8 and KC ¼ 56 is shown in Fig. 16. There is an obvious frequency peak at 1.75 Hz (7fo), the dominant frequency in this case. Contributions from 0.76 Hz (3fo), 1.0 Hz (4fo), 1.27 Hz (5fo), 1.52 Hz (6fo), 2.01 Hz (8fo) and 2.27 Hz (9fo) can also be seen but are quite insignificant. Typically, the VIV dominant frequency is governed by the St number. To further investigate the exact value of the St number, the relationship among the dominant frequency of the straked pipe, number of vortex shedding pairs, KC number under oscillatory flow, and Strouhal number can be defined as follows: N¼
fdomi fo
N fdomi Dfo fdomi D ¼ ⋅ ¼ ¼ St KC Um fo Um N ¼ St⋅KC 8
(22)
(23)
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Fig. 15. VIV response and time-frequency distribution at the 5th gauge point of the flexible pipe fitted with the helical strakes under a large KC number: (a) VR ¼ 4; (b) VR ¼ 6; (c) VR ¼ 8; (d) VR ¼ 12.
where fdomi is the dominant frequency; fo and Um are the forced oscil lation frequency and velocity amplitude, respectively. It should be noted that the diameters D in Eqs. (22) and (23) for both bare and straked pipe cases are all adopted the same value of bare pipe. The aforementioned relationship was firstly introduced to investigate the St number of a smooth rigid cylinder under oscillatory flow by (Sumer and Fredsøe, 1988). It has also been used by (Wang et al., 2015, 2016, 2017b)in the SCR and free-hanging riser model tests. By summarizing the dominant frequency for all of the test cases, the variation in the vortex shedding pairs with the KC number is shown in Fig. 17. The St number can be calculated based on Eqs. (22) and (23). Significant differences between the bare and straked pipe under oscil latory flow are observed. As shown in Fig. 17 (a), the vortex shedding pairs maintain a good linear relationship with KC numbers under VR ¼ 4, which is consistent with the results reported by (Sumer and Fredsøe, 1988). The St number is equal to 0.439 and 0.26 for the straked and bare pipes, respectively. It is unexpected that the St number of the straked pipe would be dramatically larger than that of a bare pipe, which can be attributed to the vortex-breaking effects of the helical strakes as previ ously mentioned. This finding is contrary to that under uniform flow where the helical strakes can reduce the VIV dominant frequency as reported by (Gao et al., 2014, 2015, 2016). This means that the helical strakes can increase the VIV dominant frequency under a low reduced velocity under oscillatory flow, which may lead to serious fatigue
Fig. 16. General strain amplitude-frequency spectrum for the straked pipe in the case of VR ¼ 8 and KC ¼ 56 (A ¼ 0.26 m and T ¼ 3.97 s).
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Fig. 17. Summarized relationship between the vortex-shedding pairs and KC numbers ( represents shedding pairs of the bare pipe; represents shedding pairs of is the fitting curve of the bare pipe; is the fitting curve of the straked pipe; and is the Strouhal law line). the straked pipe;
damage. When the reduced velocity increases, competition between the ef fects of the helical strakes in vortex breaking and preventing the inter action between two shear layers appears. This competition results in two distinct branches of the St numbers for the straked pipe under oscillatory flow. The main phenomena are as follows:
the first branch and increase it in the second branch. The turning point KCT is at approximately 70. (2) Similar features are also observed when the reduced velocity is equal to 8, as shown in Fig. 17 (c). The St number of the first and second branch is equal to 0.124 and 0.22, respectively. The turning point KCT is at approximately 80. (3) As the reduced velocity reaches 12, two distinct branches still exist for the straked pipe. In the first and second branches, which are divided by a KCT � 100, the St number is equal to 0.107 and 0.148, respectively. However, the St numbers of the straked pipe in these two branches are both lower than that of the bare pipe. (4) When the KC number further increases and is larger than the upper limit of that in the second branch, vortex-shedding pairs N decrease and do not maintain a linear relationship with KC number. N decreases from 40, 24, 18 at KC ¼ 140, 113, 121 to 4 at KC ¼ 165 under VR ¼ 6, 8, 12, respectively.
(1) Fig. 17 (b) shows the St number relationship when VR is 6. For the bare pipe, the St number is always equal to 0.18, which is slightly lower than the typical value of 0.2. For a straked pipe, two branches with St ¼ 0.155 (first branch) and St ¼ 0.29 (second branch) appear. The St number of the straked pipe is smaller than that of the bare pipe in the first branch with the KC number ranging from 20 to 70, while it is larger in the second branch with the KC number ranging from 70 to 140. This finding indicates that the helical strakes will decrease the VIV dominant frequency in
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The St numbers of the straked pipe in the two branches and bare pipe are summarized in Table 4. For the straked pipe, the St number de creases as the reduced velocity increases, while the turning point KCT shows an opposite tendency. This finding indicates the effect of the helical strakes in preventing the interaction between the shear layers is stronger under a higher reduced velocity due to its dependency on the Reynolds number. To summarize, the competition between the two ef fects of the helical strakes results in very complicated VIV dominant frequency for the pipe under oscillatory flow. It is possible that the he lical strakes can cause more serious fatigue damage under some condi tions, and hence, more attention should be paid to it during the engineering design process. However, under real ocean conditions, higher mode number of the flexible riser will be excited and the riser will suffer a high Re number. In our experiments, only the 1st mode was excited and it is impossible to systematically investigate the Re number effects. More experiments and numerical simulation of higher mode number and Re number effects should be conducted in the near future.
interaction between the shed vortices of the two half cycles also occurs for the straked pipe but weaker than that for the bare pipe at small KC number. Thus, lower suppression efficiency occurs at a small KC num ber. As shown in Fig. 18 (b), (c) and (d), the minimum values of the suppression efficiencies are 62.6%, 50.8% and 64.3% in the small KC number case and the maximum values are 93.5%, 91.9% and 82.4% in the large KC number case for VR ¼ 6, 8, 12, respectively. From above results, the suppression efficiencies of straked pipe at large KC number seem to be similar to that in the uniform flow. The breaking and messy vortex cannot excite a significant vibration. This may be the main rea sons for the higher suppression efficiency at larger KC number. Overall, the suppression efficiency of the helical strakes with a pitch equal to 15 D and a height equal to 0.25 D can reach 50% at a minimum. This also indicates that the suppression efficiency of other helical strake types should be verified under oscillatory flow. 4.3.2. Reduction ratio of fatigue damage Although effective suppression of the VIV of helical strakes studied in this paper can be seen from the amplitude reduction ratio, the effects of the helical strakes on fatigue damage need further study by combining them with the complicated frequency characteristics of the helical strakes under oscillatory flow. The VIV displacement amplitude reduc tion ratio cannot be considered as the unique criteria to evaluate the suppression efficiency (Resvanis et al., 2016). In our experiment, a constant pretension of 500N was applied to the ends of the model to achieve the designed natural frequencies. However, the tension force T(t) varied with time due to both varying drag force from forced motions and VIV response. Fig. 19 presents the time history of tension forced on the end of flexible pipe measured by the force transducers. The red dash line and solid line represent the tension force on straked and bare pipe, respectively. As shown in Fig. 19, compared with straked pipe, the tension force of bare pipe varies more obviously, especially for the case of larger KC number. This can be attributed to that bare pipe possess more significant VIV response than the straked pipe. The time-varying tension force can lead to a time-varying stress and contribute to the fatigue damage. Thus, the fatigue damage was derived from two parts: time-varying tension force and VIV force. To calculate the fatigue damage from measured strain signal in the model test, the bending stress and variable tension stress should be first obtained. The bending stress induced by VIV can be calculated by combing Eqs. (9) and (17), where the tension stress is obtained from the measured strain signal. As previously mentioned, the measured strain contains εT0, εT (z,t) and εVIV(z,t). According to Eqs. (7) and (8), the variable tension strain εT (z,t) can then be expressed by Eq. (25).
4.3. Suppression efficiency of helical strakes in oscillatory flow 4.3.1. Reduction ratio of VIV amplitude response The vortex-induced vibration suppression efficiency of the helical strakes, 15 D in pitch and 0.25 D in height, was tested in a rigid cylinder experiments under steady flow by Schaudt et al. (2008), obtaining at least 94% reduction. However, the suppression efficiency of a helical strake under oscillatory flow has not been studied. The efficiency η is usually defined as:
η¼
Y
Ys Y
(24)
� 100%
where Y and Ys are the standard deviation of the displacement for a bare and straked pipe, respectively. Fig. 18 shows the maximum displacement standard deviation of the VIV of bare and straked pipes. Decreasing trend of VIV response for bare pipe as KC number increases can be found in each test groups. This trend is consistent with the results found by (Wang et al., 2015). This is because the pipe encounters the previously shed vortexes as it reverses its directional motion. Such interaction is much stronger under small KC numbers, leading to a larger amplitude. This phenomenon can also be observed for a straked pipe under oscillatory flow for the same reason. Comparing the VIV response between the bare and straked pipes, the helical strake type can still effectively suppress the VIV under oscillatory flow as expected. However, the suppression efficiency is not as high as that under uniform flow as reported by (Schaudt et al., 2008). Under a lower reduced velocity, the broken shedding vortexes of the straked pipe are quite weak and cannot excite a significant VIV. Under this condition, the VIV displacements are stable with the KC number as shown in Fig. 16 (a). While the interaction between the shed vortices of the two half cycles for the bare pipe is stronger, the VIV response decreases as the KC number increases. Therefore, the suppression efficiency can reach 92.7% for a small KC number, but decreases to 71.9% for a large KC number. However, for a higher reduced velocity (VR ¼ 6, 8 and 12), the
εT ðz; tÞ ¼ f½εCF a ðz; tÞ
Straked pipe First branch
4 6 8 12
Bare pipe KCT
St
KC
0.439 0.155 0.124 0.107
– 20–70 20–80 20–100
– 70 80 100
Second branch St
KC
0.439 0.29 0.22 0.148
– 70–140 80–120 100–120
εT0 �g=2
(25)
In the Eq. (25), the initial tension strain εT0 is obtained through averaging the initial strain as shown in Fig. 20. Combing Eqs. (25) and (17), the variable tension stress can be calculated. Fig. 21 presents the distributions of root mean square of stress caused by VIV and varying tension force under KC ¼ 30 and 121 for VR ¼ 6.0. The solid line and dash line represent stress of bare and straked pipe, respectively. The blue line and red line represent the case of KC ¼ 30 and 121, respectively. As shown in Fig. 21 (a), the more obvious VIV response of bare pipe pos sesses more significant VIV stress σVIV_RMS than the straked one. σVIV_RMS at the middle points is obviously larger than those close to the end of the model, while the tension stress σT_RMS distributes evenly along the pipe as shown in Fig. 21 (b). Compared with σVIV_RMS, the stress derived from variable tension force is relatively smaller and there are no obvious differences under different KC numbers. After obtaining the stress, the stress cycles can be identified from the time history by the rainflow procedure (Amzallag et al., 1994). Then, according to Eq. (20), fatigue damage derived from VIV and varying tension force at each gauge point can be calculated based on Eqs. (18–20), respectively. Fig. 22 shows the fatigue damage distributions along the flexible pipe under KC ¼ 30 and 121 for VR ¼ 6. As shown in
Table 4 Strouhal number and KC number region for the straked and bare pipes. VR
εT0 � þ ½εCF c ðz; tÞ
St 0.26 0.18 0.146 0.174
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Ocean Engineering 189 (2019) 106274
Fig. 18. Maximum displacement standard deviation of the VIV for the bare and straked pipes ( is the displacement response of the bare pipe; response of the straked pipe).
Fig. 22 (a), it can be seen that the fatigue damage caused by VIV bending strain Dot_VIV at the middle location is larger due to more obvious VIV response at this location. Dot_VIV of bare pipe is significant larger than that of straked one. It indicates that the helical strakes can reduce the Dot_VIV and prolong service life of the riser in an oscillatory flow. Fig. 22 (b) shows the distribution of fatigue damage derived from the varying tension force Dot_T. Compared with Dot_VIV, the Dot_T distributes more evenly along the flexible pipe and very smaller. Dot_T is about 104 orders of magnitude smaller than Dot_VIV. It means fatigue damage from vari able tension force can be ignored in our experiments. Beyond that, the variable tension force is not only caused by the VIV, it also affected by varying drag force from forced motions. To investigate the effects of helical strakes on reducing fatigue damage caused by pure VIV response, the maximum fatigue damage of Dot_VIV in each case is summarized. Fig. 23 shows the variation in the maximum fatigue damage of Dot_VIV with KC number for a bare and straked pipe. The maximum fa tigue damage decreases as the KC number increases for the bare pipe. The reason is the same as discussion of the VIV response amplitude distribution shown in Fig. 18.
is the displacement
Compared to the fatigue damage of the bare pipe, we surprisingly found that the maximum fatigue damage of the straked pipe under a low reduced velocity VR ¼ 4 is close to that of the bare pipe at a large KC number. Combining the Strouhal relationship shown in Fig. 17 and response amplitude distribution shown in Fig. 18, a higher VIV domi nant frequency caused by the vortex-breaking effect of the helical strakes may be the main reason for this phenomenon. As shown in Fig. 23 (a), the fatigue damage reduction rate is over 95%, while it de creases to 41% at a KC ¼ 121. When the reduced velocity further increases to 6, 8 and 12 as shown in Fig. 23 (b), (c), and (d), respectively, a similar trend of VIV response amplitude distribution is also observed in the maximum fatigue damage. The fatigue damage of the straked pipe under a small KC number is not as ideal as that under a large KC number. Although the dominant fre quency of the straked pipe at the second branch as previously mentioned is higher than that of the bare pipe, the response amplitude is effectively reduced at this branch. Thus, the effects of the two branches on the Strouhal number caused by covering the helical strakes cannot be seen in the maximum fatigue damage distribution. Due to limitation of the 12
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Fig. 19. Time history of tension force on the ends of the straked and bare pipe under KC ¼ 30 and KC ¼ 121 for VR ¼ 6.0.
Fig. 20. Time history of εCF_a (z,t) and εCF_c (z,t) of the straked pipe under VR ¼ 6.0 and KC ¼ 30.
Fig. 22. Fatigue damage distributions of the straked and bare pipe caused by VIV and varying tension force under KC ¼ 30 and KC ¼ 121 for VR ¼ 6.0.
5. Conclusions In this paper, an experimental study of a flexible pipe fitted with helical strakes was conducted in oscillatory flow under a KC number ranging from 20 to 160 and a maximum reduced velocity varying from 4 to 12. Compared with bare pipe, the effects of the helical strakes on the VIV response, St number, suppression efficiency and fatigue damage under oscillatory flow were investigated. The main conclusions are as follows:
Fig. 21. Stress distributions of the straked and bare pipe caused by VIV and varying tension force under KC ¼ 30 and KC ¼ 121 for VR ¼ 6.0.
(1) Under a lower reduced velocity, the “mode transition” phenom enon for the straked pipe is quite obvious, and the helical strakes can significantly increase the VIV dominant frequency. The Strouhal number of the straked pipe can reach 0.439, which is nearly twice that of the typical St number measured during a steady flow experiment. (2) Under a higher reduced velocity, two distinct branches in the variation of the St number against the KC number are observed. The helical strake reduces the dominant frequency in the first
length of the flexible pipe, higher modes were not excited in our ex periments. In the future, more work to excite higher mode of a flexible pipe in an oscillatory flow will be conducted. Overall, the fatigue damage for the straked pipe under oscillatory flow under a low reduced velocity and small KC number is larger and should be paid more atten tion in practical engineering projects.
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Fig. 23. Maximum fatigue damage distribution with the KC number of the bare and straked pipe ( is the fatigue damage of the bare pipe and of the straked pipe).
branch with a small KC number, while it increases it in the second branch with a large KC number. (3) The helical strake shows its effects in suppressing VIV under oscillatory flow. However, the suppression efficiency of the VIV and fatigue damage, particularly under the conditions of a small KC number and low reduced velocity, is not as high as that under steady flow. More attentions should be paid to this phenomenon in practical engineering projects.
is the fatigue damage
Acknowledgments The authors gratefully acknowledge the support from the National Science Fund for Distinguished Young Scholars of China (No. 51825903), Shanghai Science and Technology Program (No. 19XD1402000), the National Natural Science Foundation of China (No. 51490674), National Program on Key Basic Research Project of China 14
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(973 Program) (No. 2015CB251203) and Ub-project of the Important National Science & Technology Specific Projects of China (No. 2016ZX05028-002-004). And all authors also would like to acknowl edge permission and support from SBM Offshore to prepare and publish this work.
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